MISN-0-35
MOMENTS OF INERTIA,
PRINCIPAL MOMENTS
MOMENTS OF INERTIA, PRINCIPAL MOMENTS
by
J. S. Kovacs
I A =I B
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. L and ω for a Rotating Rigid Object; Principal Axes of
Inertia
a. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
b. Additional Study Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
A
3. Moment of Inertia by Two Methods . . . . . . . . . . . . . . . . . . . 3
B
4. The Radius of Gyration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
Project PHYSNET · Physics Bldg. · Michigan State University · East Lansing, MI
1
ID Sheet: MISN-0-35
THIS IS A DEVELOPMENTAL-STAGE PUBLICATION
OF PROJECT PHYSNET
Title: MOMENTS OF INERTIA, PRINCIPAL MOMENTS
Author: J. S. Kovacs, Michigan State University
Version: 2/1/2000
Evaluation: Stage 6
Length: 1 hr; 16 pages
Input Skills:
1. Use the scalar product to determine the angle between two given
vectors (MISN-0-2).
2. Set up and solve integrals in one, two, and three dimensions
(MISN-0-1).
3. State the law of conservation of angular momentum (MISN-0-41).
Output Skills (Knowledge):
K1. Define moment of inertia, principal axes of inertia, and principal
moments of inertia.
K2. Define the radius of gyration of a rigid body.
Output Skills (Project):
P1. Given the angular velocity and the angular momentum vectors of
a rotating rigid body with respect to some fixed reference system,
determine whether or not the axis of rotation is one of the principal
axes of the body. Also determine the moment of inertia of this
object with respect to this rotation axis.
P2. Calculate the moment of inertia relative to a given axis of simple
mass configurations when the mass and the distribution of mass
are given. Also, given the radius of gyration, determine the moment of inertia of a rigid body.
P3. Use the conservation of angular momentum principle to determine the moment of inertia of an irregularly shaped object when
a change in location of a part of the object of known moment of
inertia produces an observed change in angular velocity.
The goal of our project is to assist a network of educators and scientists in
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processing and distribution, along with communication and information
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as well as physics topics that are needed in science and technology. A
number of our publications are aimed at assisting users in acquiring such
skills.
Our publications are designed: (i) to be updated quickly in response to
field tests and new scientific developments; (ii) to be used in both classroom and professional settings; (iii) to show the prerequisite dependencies existing among the various chunks of physics knowledge and skill,
as a guide both to mental organization and to use of the materials; and
(iv) to be adapted quickly to specific user needs ranging from single-skill
instruction to complete custom textbooks.
New authors, reviewers and field testers are welcome.
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Webmaster
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Project Director
ADVISORY COMMITTEE
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Views expressed in a module are those of the module author(s) and are
not necessarily those of other project participants.
External Resources (Required):
1. M. Alonso and E. J. Finn, Physics, Addison-Wesley, Reading
(1970). For availability, see this module’s Local Guide.
c 2001, Peter Signell for Project PHYSNET, Physics-Astronomy Bldg.,
°
Mich. State Univ., E. Lansing, MI 48824; (517) 355-3784. For our liberal
use policies see:
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MISN-0-35
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MISN-0-35
~ in the same direction as ω
~ . That is,
where Lω is the component of L
MOMENTS OF INERTIA,
PRINCIPAL MOMENTS
~ ·ω
L
~ = Lω cos α ≡ Lω ω,
by
J. S. Kovacs
1. Introduction
In general, the motion of a rigid body consists of a translation
(straight line or some other trajectory motion) of the center of mass plus a
rotation of the object about some axis through the center of mass. In this
unit we treat the second part alone, the case of rotation only. Therein,
we develop the relationship between the angular momentum and angular velocity vectors. If, for some rotation, these two vectors are parallel,
the axis of rotation is said to be a “principal axis.” Involved in these
relationships is the concept of moments of inertia. We consider how to
calculate the moment of inertia of a given object from the definition of
that quantity.
2. L and ω for a Rotating Rigid Object; Principal
Axes of Inertia
2a. Discussion. A point mass moving along a circular path has an
angular velocity vector, ω
~ , directed along the axis of the circle, and an
~ , relative to the center of the circle which is
angular momentum vector, L
parallel to the angular velocity. The quantities are related in magnitude
by L = M R2 ω where M is the particle mass and R is the radius of the
circle. The combination M R2 is the moment of inertia of the point mass
relative to the axis of rotation.1 An extended rigid body may be viewed
as a distribution of point masses. If such a rigid body rotates about
some fixed axis, the angular velocity vector and the angular momentum
vector are not, in general, parallel. However, a relation between ω
~ and
~ which is parallel to ω
the component of L
~ can still be written down. The
proportionality factor is the moment of inertia of the rigid body relative
to the axis of rotation,
Lω = Iω ω,
(1)
1 These quantities are defined and related in “Torque and Angular Acceleration
for Rigid Planar Objects: Flywheels” (MISN-0-33). See also “Kinematics: Circular
Motion” (MISN-0-9).
5
~ and ω
α being the angle between L
~ . I is the moment of inertia of the
rigid body relative to the axis of rotation determined by the vector ω
~ . For
an extended rigid object it is the analog of what M R2 is for a point object
of mass M moving in a circle of radius R. In fact, by considering the rigid
body as a collection of point masses, each its own distance from the axis
of rotation, one can directly calculate the moment of inertia for the object
~ (those perpendicular
under consideration.2 The other components of L
to ω
~ ) cannot be related to ω
~ via a relation such as Eq. (1).
~ is parallel to ω
~ has no components perpenSuppose that L
~ . Then L
dicular to ω
~ , and Eq. (1) can be written as a vector equation:
~ = I~
L
ω.
(2)
The conditions under which Eq. (2) is satisfied usually exist if the angular
velocity ω
~ is directed along one of the symmetry axes of the object.3 These
are called the principal axes of inertia of the object and the moments of
inertia relative to these axes are the principal moments of inertia.
As an illustration of these concepts, consider the following situation.
Suppose that relative to a fixed inertial reference frame the angular momentum vector of a rotating rigid body is given by:
~ = [A sin(α) sin(ωt + δ)]x̂ + [A sin(α) cos(ωt + δ)]ŷ + [A cos α]ẑ.
L
Here A, α and δ are constants, t is the time, and ω is the magnitude of the
angular velocity of the rotating rigid body. The vector ω
~ has magnitude
and direction given by ω
~ = ωẑ. The quantities x̂, ŷ, and ẑ are unit vectors
along the coordinate axes of the inertial reference frame. It is clear that
~ does).
~ and ω
L
~ are not parallel (~
ω does not have x- and y-components, L
The axis of rotation of this object is thus not one of the principal axes of
inertia.
~ is A, the angle between L
~
It can be verified that the magnitude of L
and ω
~ is α, and that the moment of inertia of this object relative to the
z-direction is (A/ω) cos α. No other moment of inertia for this body can
be determined from the information supplied.
2 This
calculation is the subject of the next section of this module.
even if an object has no symmetry associated with it, it can be proven
that there are at least three mutually perpendicular directions for which Eq. (2) is
satisfied if the rotation is about one of these axes.
3 However,
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MISN-0-35
2b. Additional Study Material. In AF4 study Sections 11.1 and
11.2 (p. 210-213), especially Example 11.1 and Fig. 11.6.
3. Moment of Inertia by Two Methods
Learn to get the moment of inertia: (1) by direct calculation; and
(2) by conservation of angular momentum.
LG-1
MISN-0-35
LOCAL GUIDE
The readings for this unit are on reserve for you in the Physics-Astronomy
Library, Room 230 in the Physics-Astronomy Building. Ask for them as
“The readings for CBI Unit 35.” Do not ask for them by book title.
Study: In AF, Section 11.3 (pp. 213-217), Section 10.5 (pp. 188189). For access, see this module’s Local Guide.
4. The Radius of Gyration
If all of the mass of an object were located at a point instead of being
distributed as it is, where should all of this mass be so that the moment
of inertia of the mass relative to some axis is the same as for the object
itself? The distance from the axis to this point is called the radius of
gyration, and is usually designated by k. The moment of inertia of the
object is related to the radius of gyration by:
I = M k2 ,
as defined for a point mass M .
As an example, the radius of gyration of a sphere of radius R relative
to an axis through its center is:
√
k = 0.4R ' 0.632R.
Acknowledgments
Preparation of this module was supported in part by the National
Science Foundation, Division of Science Education Development and Research, through Grant #SED 74-20088 to Michigan State University.
4 M. Alonso and E. J. Finn, Physics, Addison-Wesley, Reading (1970). For access,
see this module’s Local Guide.
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PS-1
MISN-0-35
PS-2
MISN-0-35
`
w
Irregularly Shaped Object
PROBLEM SUPPLEMENT
1. For Fig. 11.6 in AF5 verify:
~ is A.
a. The magnitude of L
~ and ω
b. The angle between L
~ is α.
A
c. The moment of inertia of the object with respect to the z-axis is
numerically equal to (A/ω) sin α.
2. For a rotating rigid body the angular velocity it has relative to a fixed
coordinate system is given by ω
~ = ωx x̂ + ωy ŷ, where ωx and ωy are
constants. Its angular momentum relative to this same set of axes is
~ = Lx x̂ + Ly ŷ. With Lx written as Aωx and Ly written as
given by: L
Bωy , determine the following:
a. Is the axis of rotation of this object “fixed”?
~ and ω
b. Are L
~ parallel?
c. What is the moment of inertia of this rigid body with respect to
the given axis of rotation?
d. Is the axis about which the rotation is taking place a principal axis
of rotation for the object?
e. Under what condition (expressed in terms of the coefficients A and
B) will this axis be a principal axis for the object?
f. What is then the principal moment of inertia relative to this axis?
3. How would you determine the moment of inertia of an irregularly
shaped object such as, say, your own body? You can’t calculate it
by direct integration as you would, for example, for a cylinder, relative
to its axis. You can, however, use the principle of angular momentum
conservation to determine it experimentally.
Consider that you wish to determine your moment of inertia relative to
a vertical axis through the center of mass of your body (when you are
in a standing position). If you stand on a rotating turntable rotating
at a constant angular speed about a vertical axis, the component of
your angular moment along the axis of rotation is proportional to the
moment of inertia you want to determine.
B
M
x
M
x
Massless Rod
Fixed to Object
0.6m
0.6m
Turntable
Figure 1. Irregularly shaped object rotating about vertical
axis on a turntable which rotates with very little frictional
loss. Two masses M are located on a massless rod held
horizontally fixed to the object.
The problem is to extract it from the angular momentum which is also
unknown. This can be done by causing a known change in the moment of inertia of the system (you and anything you might be holding)
and observing the change in angular velocity. For example, Suppose
you are holding a long rod of negligible mass horizontally across your
body. Attached to this rod and free to slide along the rod are two 4
kilogram masses (small enough to be treated as point-like), symmetrically placed on the rod at opposite sides of your body each 0.6 meters
from the vertical axis through your body (See Figure 1). The masses
are temporarily held fixed at the 0.6 meter location. The system (you
and the loaded rod) is observed to rotate at 30 revolutions per minute.
At some instant the masses are released and move out to point A and
B in Fig. 1, each point being 1.2 m from the vertical axis. With this
rearrangement the system is observed to now rotate at 10 revolutions
per minute. Determine your moment of inertia from this data.
5 M. Alonso and E. J. Finn, Physics, Addison-Wesley, Reading (1970). For access,
see this module’s Local Guide.
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MISN-0-35
PS-3
ME-1
MISN-0-35
4. In AF work problems 11.1, 11.2, 11.3 and 11.4.
5. a. For the person on the turntable described in problem 3, determine
the radius of gyration with respect to the vertical axis if the person
weighs 160 pounds (determine the radius of gyration in meters).
b. Determine the radius of gyration of the person plus masses-on-therod system when the masses are at points A and B.
Note: This Model Exam consists of more detailed step-by-step questions
than the actual exam is likely to have. Consider it both as a Model Exam
and a Review.
1. See Output Skills K1-K2 on this module’s ID Sheet. One or more of
these skills, or none, may be on the actual exam.
Brief Answers:
2. Relative to the principal axes of the body (call the direction of the
axes x̂, ŷ, and ẑ, the principal moments of inertia Ix , Iy , and Iz . If
the body is rotating with an angular velocity ω = −ω0 x̂ (where ω0 is
a positive number), what is the axis about which the rotation of this
object occurs? [K] Relative to this axis, which way does this object
rotate? [R] What angular momentum does this object have? [M]
2. a. Yes, ω
~ is constant.
b. Not in general. Depends on values of A and B.
c.
MODEL EXAM
Aωx2 + Bωy2
.
ωx2 + ωy2
~ and ω
d. No, L
~ aren’t parallel for arbitrary A and B.
~ = A~
e. If A = B then L
ω and they are parallel.
3. At a certain instant, the angular momentum vector of an object relative
to fixed cartesian coordinate system is given by:
~ = (2x̂ + 7ŷ − 5ẑ) kg m2 /s.
L
f. A.
while its angular velocity vector is given by:
3. 1.44 kg m2 .
ω
~ = (8x̂ + 4ŷ) rad/s.
4. AF 11.1 Radius of gyration, when axis is at one end, is 0.612 m.
AF 11.2 a. I = 1.94 kg m2 , k = 0.611m.
a. Is this system rotating about one of the body’s principal axes? [E]
Explain. [V]
AF 11.2 b. Irod = 48M L2 where M = 0.20 kg is the total mass of the
rod and L = 1.0 m is the length of the rod. For the system, rod plus
masses, Itotal = O.966 kg m2 , k = 0.431 m.
b. What is the moment of inertia of this body with respect to the x-,
y-, and z-axes? [H]
~ [P]
c. What is the magnitude of L?
AF 11.2 c. Itotal = 0.642 kg m2 , k = 0.351 m.
d. What is the magnitude of ω? [F]
AF 11.3 Answers in AF are correct. For part (c), recall that the center
of mass of the equilateral triangle is at the intersection of the medians
(see Table 4.1, p. 47) which is two-thirds of the distance from either
vertex to the midpoint of the opposite side.
5. Answers: (a) 0.14 m; (b) 0.4 m.
e. What is the moment of inertia of this body with respect to the axis
determined by the direction of ω
~ ? [B]
4. A right circular cylinder of uniform volume density ρ, length L, and radius R, has a total mass
M . It is shown in cross-section in the sketch
(the length L is perpendicular to the page).
dr
r
R
11
12
ME-2
MISN-0-35
a. Express the density ρ in terms of M , L, and R. [J]
b. How much mass is in a cylindrical shell of thickness dr (shown
shaded in sketch), located at a distance r from the axis of the cylinder? [T]
MISN-0-35
ME-3
f. What is the moment of inertia relative to an axis through the center
of mass? (Use the parallel axis theorem.) [I]
g. What is the moment of inertia relative to an axis through the x = L
end? (Get the answer using the parallel axis theorem and by direct
integration.) [W]
c. What is the moment of inertia, relative to the axis of the cylinder,
of the mass contained in this cylindrical shell of thickness dr? [N]
d. Sum up, by integration, the contributions to the total moment of
inertia from all of the cylindrical shells from the axis out to radius
R. [U]
e. Eliminate ρ from above to get I in terms of M and R. [A]
f. What is the radius of gyration of this cylinder relative to this axis?
[S]
5.
dx
0
x
L
A nonuniform rod of length L has a density that increases linearly from
the end at x = 0 to the end x = L. The density (mass per unit length)
is given by ρ = bx, where b is a constant and x is the distance from
the end. The density at the x = 0 end is zero.
a. At a distance x from the x = 0 end, what is the mass of an increment
of length dx? [G]
b. Sum up the contributions to the total mass from all of the elements
of length dx to find the total mass of the rod (in terms of b and L).
[O]
c. Where is the center of mass of this rod? If you can’t recall the
definition of this, refer to a text. [Q]
d. What is the moment of inertia, relative to an axis through x = 0,
of the element of mass located in thickness dx at distance x from
the x = 0 end? [X]
e. What is the total moment of inertia of this rod relative to an axis
through the x = 0 end? [L] What is the radius of gyration of this
rod relative to the x = 0 end? [Y]
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MISN-0-35
ME-4
Brief Answers:
A. M R2 /2.
B. 0.55 kg m2 .
E. No.
√
F. 80 rad/s= 8.94 rad/sec.
G. bx dx.
H. Cannot be determined from information given.
I. Ic = M L2 /18.
J. M/(πr 2 L) − 1.
K. The x-axis.
L. M L2 /2.
M. Ix ω0 , in the same direction as ω
~.
N. 2πρLr 3 dr.
O. M = −bL2 /2, so b = 2M/L2 .
√
P. 78 kg m2 /s.
Q. Xc = 2L/3.
R. Use the right-hand rule: if thumb is in direction of ω, sense of rotation
is in direction of curled fingers.
√
S. 0.5R2 = 0.707 R.
T. 2ρLπrdr.
U.
1
πρL4 .
2
~ and ω
V. Because L
~ are not in the same direction.
W. I = 13/8M L2.
X. dI = bx3 dx.
Y. k = 0.707L.
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